Transcript Section 6.4 – Logarithmic Functions

Section 6.3 – Exponential Functions
Laws of Exponents
If s, t, a, and b are real numbers where a > 0 and b > 0, then:
𝑎 𝑠 ∙ 𝑎𝑡 = 𝑎 𝑠+𝑡
𝑡
1 =1
(𝑎 𝑠 )𝑡 = 𝑎 𝑠𝑡
(𝑎𝑏)𝑡 = 𝑎𝑡 𝑏 𝑡
0
𝑎 =1
𝑎−𝑡
1
1
= 𝑡=
𝑎
𝑎
Definition:
An Exponential Function is in the form, 𝑓 𝑥 = 𝐶𝑎 𝑥
“a” is a positive real number and does not equal 1
“a” is the base and is the Growth Factor
“C” is a real number and does not equal 0
“C” is the Initial Value because 𝑓 0 = 𝐶𝑎0 = 𝐶
The domain of f(x) is the set of all real numbers
𝑓(𝑥 + 1)
𝐶𝑎 𝑥+1 𝐶𝑎 𝑥 𝑎1
=𝑎
→
=
=𝑎
𝑥
𝑥
𝑓(𝑥)
𝐶𝑎
𝐶𝑎
𝑡
Section 6.3 – Exponential Functions
Examples
𝑥
𝑓(𝑥)
𝑓(𝑥 + 1)
=𝑎
𝑓(𝑥)
−1
2
3
1
3
=
2/3 2
0
1
3/2 3
=
1
2
1
3
2
9/4 3
=
3/2 2
2
9
4
27/8 3
=
9/4
2
3
27
8
𝑎=
𝑓 𝑥 = 𝐶𝑎 𝑥
𝑓 0 = 1, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐶 = 1
3
𝑓 𝑥 =1
2
𝑥
3
2
3
=
2
𝑥
Section 6.3 – Exponential Functions
Examples
𝑥
𝑓(𝑥)
𝑓(𝑥 + 1)
=𝑎
𝑓(𝑥)
−1
1
2
1/4 1
=
1/2 2
0
1
4
1/8 1
=
1/4 2
1
1
8
1/16 1
=
1/8
2
2
1
16
1/32 1
=
1/16 2
3
1
32
𝑎=
𝑓 0 = 1/4, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐶 = 1/4
1 3
𝑓 𝑥 =
4 2
𝑥
1
2
𝑓 𝑥 = 𝐶𝑎 𝑥
Section 6.3 – Exponential Functions
Properties of the Exponential Function,𝑓 𝑥 = 𝑎 𝑥 , 𝑎 > 1
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.
If a > 1, the f(x) is increasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 𝑥 = 𝑎𝑥 , 𝑎 > 1
The graph of the exponential function is shown below.
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
Section 6.3 – Exponential Functions
Properties of the Exponential Function, 𝑓 𝑥 = 𝑎 𝑥 , 0 < 𝑎 < 1
The domain is the set of all real numbers.
The range is the set of all positive real numbers.
The y-intercept is 1; x-intercepts do not exist.
The x-axis (y = 0) is a horizontal asymptote, as x.
If 0 < a < 1, then f(x) is a decreasing function.
The graph contains the points (0, 1), (1, a), and (-1, 1/a).
The graph is smooth and continuous.
Section 6.3 – Exponential Functions
𝑓 𝑥 = 𝑎𝑥 , 0 < 𝑎 < 1
The graph of the exponential function is shown below.
𝑓 𝑥 = 𝑎𝑥 , 0 < 𝑎 < 1
Section 6.3 – Exponential Functions
Euler’s Constant – e
The value of the following expression approaches e,
1
1+
𝑛
𝑛
as n approaches .
Using calculus notation,
1
𝑒 = lim 1 +
𝑛→∞
𝑛
𝑛
Applications of e
Growth and decay
Compound interest
Differential and Integral calculus with exponential functions
Infinite series
Section 6.3 – Exponential Functions
Theorem
If 𝑎𝑢 = 𝑎𝑣 , then 𝑢 = 𝑣.
Solving Exponential Equations
1) 32𝑥−1 = 3𝑥
2𝑥 − 1 = 𝑥
x=1
2) 22𝑥 = 32
22𝑥 = 4 ∙ 8
3) 4𝑥+2 = 8
22
𝑥+2
= 23
22𝑥 = 22 ∙ 23
22(𝑥+2) = 23
22𝑥 = 25
2𝑥 = 5
5
𝑥=
2
22𝑥+4 = 23
2𝑥 + 4 = 3
1
𝑥=−
2
Section 6.3 – Exponential Functions
Solving Exponential Equations
4)
1 𝑥+2
3
3−1
93𝑥
=
𝑥+2
= 32
3−𝑥−2 = 36𝑥
−𝑥 − 2 = 6𝑥
𝑥=−
2
7
3𝑥
5) 81 ∙ 9−2𝑏−2 = 27
9 ∙ 9 ∙ 9−2𝑏−2 = 27
32 ∙ 32 32 −2𝑏−2 = 33
34 ∙ 3−4𝑥−4 = 33
3−4𝑥 = 33
−4𝑥 = 3
3
𝑥=−
4
Section 6.4 – Logarithmic Functions
The exponential and logarithmic functions are inverses of each other.
The logarithmic function is defined by
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
𝑥 = 𝑎𝑦
The domain is the set of all positive real numbers 0, ∞ .
The range is the set of all real numbers −∞, ∞ .
The x-intercept is 1 and the y-intercept does not exist.
The y-axis (x = 0) is a vertical asymptote.
If 0 < a < 1, then the logarithmic function is a decreasing function.
If a > 1, then the logarithmic function is an increasing function.
The graph contains the points (1, 0), (a, 1), and (1/a, –1).
The graph is smooth and continuous.
Section 6.4 – Logarithmic Functions
The graph of the logarithmic function is shown below.
𝑎, 1
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
𝑎
The natural logarithmic function
𝑦 = 𝑙𝑜𝑔𝑒 𝑥 = ln 𝑥
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
𝑥 = 𝑒𝑦
𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
𝑥 = 10𝑦
The common logarithmic function
𝑦 = 𝑙𝑜𝑔10 𝑥 = 𝑙𝑜𝑔 𝑥
Section 6.4 – Logarithmic Functions
Graphs of 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = 𝑎 𝑥
𝑎, 1
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
𝑎
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
Section 6.4 – Logarithmic Functions
Graphs of 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = 𝑎 𝑥
Inverse Functions: 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = 𝑎 𝑥
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1
𝑎
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
Section 6.4 – Logarithmic Functions
Graphs of 𝑓 𝑥 = 𝑒 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = ln 𝑥
Inverse Functions: 𝑓 𝑥 = 𝑒 𝑥 𝑎𝑛𝑑 𝑓 𝑥 = ln 𝑥
Section 6.4 – Logarithmic Functions
Graph 𝑓 𝑥 = 𝑎 𝑥 + 2
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1
𝑎
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
Section 6.4 – Logarithmic Functions
Graph 𝑓 𝑥 = 𝑎 𝑥+1 + 2
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1
𝑎
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
Section 6.4 – Logarithmic Functions
Graph 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥 − 1
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1

𝑎
𝑦 = 𝑙𝑜𝑔𝑎 𝑥
Section 6.4 – Logarithmic Functions
Graph 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 (𝑥 − 2) − 1
𝑦 = 𝑓 𝑥 = 𝑎𝑥
a
𝑎, 1
𝑎
𝑦 = 𝑙𝑜𝑔𝑎 𝑥

Section 6.4 – Logarithmic Functions
Change the exponential statements to logarithmic statements
𝑎5 = 6.7
8𝑥 = 9.2
𝑒3 = 𝑏
𝑒𝑥 = 4
5 = 𝑙𝑜𝑔𝑎 6.7
𝑥 = 𝑙𝑜𝑔8 6.7
3 = ln 𝑏
𝑥 = ln 4
Change the logarithmic statements to exponential statements
𝑙𝑜𝑔3 5 = 𝑥
𝑙𝑜𝑔𝑥 7 = 4
ln 𝑎 = 6
5 = 3𝑥
7 = 𝑥4
𝑎 = 𝑒6
Solve the following equations
𝑙𝑜𝑔3 (2𝑥) = 1
𝑙𝑜𝑔2 (5𝑥 + 1) = 4
𝑙𝑜𝑔2 32 = −3𝑥 + 9
2𝑥 = 31
3
𝑥=
2
5𝑥 + 1 = 24
32 = 2−3𝑥+9
5𝑥 + 1 = 16
25 = 2−3𝑥+9
5𝑥 = 15
𝑥=3
5 = −3𝑥 + 9
−4 = −3𝑥
4
=𝑥
3
Section 6.4 – Logarithmic Functions
Solve the following equations
𝑒 𝑥 = 10
𝑒 7𝑥 = 15
8 + 2𝑒 𝑥 = 12
𝑥 = ln 10
7𝑥 = ln 15
ln 15
𝑥=
7
2𝑒 𝑥 = 4
𝑒𝑥 = 2
𝑥 = ln 2